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3 results

  • 3 - Quadratic Forms

  • John Stillwell, University of San Francisco
  • Book:
    Algebraic Number Theory for Beginners
    Published online:
    28 July 2022
    Print publication:
    11 August 2022, pp 59-77

  • Summary

    Solving quadratic Diophantine equations amounts to finding the values taken by quadratic forms, a problem that can be fruitfully approached by finding the equivalents of a given form under change of variables. This approach was initiated by Lagrange and developed to a high level by Gauss. However, the way Gauss did it involved an apparently difficult operation called composition of forms, clarified only later by the concept of Abelian group.

  • 2 - Diophantine Arithmetic

  • John Stillwell, University of San Francisco
  • Book:
    Algebraic Number Theory for Beginners
    Published online:
    28 July 2022
    Print publication:
    11 August 2022, pp 33-58

  • Summary

    Diophantine equations are polynomial equations for which integer (or sometimes rational) solutions are sought. The oldest examples date from ancient Greek times, and Diophantus in particular solved many such equations. His methods and the questions they raised inspired much of modern number theory, beginning with the work of Fermat and Euler. Euler, and later Gauss, introduced algebraic integers to solve Diophantine equations, implicitly or explicitly using "unique prime factorization" to do so.